The pressure of water waves upon a fixed obstacle 415 
TABLE 1 
NG 0° 45° 90° I1BES 180° 
Ka 
0-5 1:44 1:28 0:97 0-91 1:00 
1-0 1:71 1-62 1-16 0-68 0-82 
3-0 1:92 1:75 1:35 0-82 0-62 
5:0 1:96 1-86 1:36 0-64 0:48 
5. For the parabolic cylinder we may use the expressions given by Lamb 
(1906) for the diffraction of sound waves, making the necessary modifica- 
tions for water waves. In this case we take the plane waves to be moving 
in the positive direction of Ox; the water-plane section of the cylinder 
is given by 
Key? = 4) 4+ 4kyeu. (19) 
In the parabolic co-ordinates defined by «(a+iy) = (€+7”)?, the section 
of the cylinder is given by 7 = 7p. 
The velocity potential of the motion is 
db = (gh/c) ciel - ace ay, (20) 
7 
where the constant C is given by 
2ino|I £6 | eee a — Cetin? = 0. (21) 
For the surface elevation on the cylinder, we have 
‘ae inet +0 | * en2ie al : (22) 
0 
It follows that the amplitude of the oscillation is constant round the 
boundary. From (21) and (22) we find that this amplitude is h/p, where p 
is given by 
p? = 1+ 2ming{(} —c) sin 293 —(—s) cos 23} + mm8{(—c)? + (4—s)?}, (23) 
in which ¢ and s denote Fresnel integrals of argument 27)7-?. 
From (19) we have 27? = xa, where a is the radius of curvature of the 
parabola at its vertex. Table 2 gives the ratio of the amplitude of the 
oscillation to that of the incident waves, calculated from (23) for certain 
values of Ka: 
TABLE 2 
Ka 0:05 0-1 1:0 3-0 5-0 
amp./h. 1-28 1-40 1-65 1-86 1-93 
476 
